A reasonable physical intuition in the study of interacting quantum systems says that, independent of the initial state, the system will tend to equilibrate. In this work we study a setting where relaxation to a steady state is exact, namely for the Bose-Hubbard model where the system is quenched from a Mott quantum phase to the strong superfluid regime. We find that the evolving state locally relaxes to a steady state with maximum entropy constrained by second moments, maximizing the entanglement, to a state which is different from the thermal state of the new Hamiltonian. Remarkably, in the infinite system limit this relaxation is true for all large times, and no time average is necessary. For large but finite system size we give a time interval for which the system locally "looks relaxed" up to a prescribed error. Our argument includes a central limit theorem for harmonic systems and exploits the finite speed of sound. Additionally, we show that for all periodic initial configurations, reminiscent of charge density waves, the system relaxes locally. We sketch experimentally accessible signatures in optical lattices as well as implications for the foundations of quantum statistical mechanics.

This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $\pi / 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and $\pi / 8$ gates. The algorithm runs in $O(\log^{2.71}(1/\epsilon))$ time, and produces as output a sequence of $O(\log^{3.97}(1/\epsilon))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $\epsilon > 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d).

We consider the implementation of two-qubit unitary transformations by means of CNOT gates and single-qubit unitary gates. We show, by means of an explicit quantum circuit, that together with local gates three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates.PACS numbers: 03.67.Mn, 03.67.Lx In the context of establishing the existence of universal sets of two-qubit gates for quantum computation [1], Barenco et al. [2] showed that any unitary transformation on n qubits can be decomposed into a sequence of CNOT and single-qubit gates. Since then it has become customary to express n-qubit unitary transformationsassociated e.g. with quantum algorithms-as a series of CNOT and single-qubit gates in the quantum circuit model [3].Relatedly, in those experimental settings where significant single-qubit control is already available, the ability to reliably perform a CNOT gate has become the standard hallmark of multi-qubit control. As a consequence, achieving a CNOT gate is one of the most popular and coveted goals among quantum information experimentalists [4]. In turn, this has triggered theoretical studies on the optimal use of two-qubit interactions and of entangling gates to perform a CNOT gate [5,6,7,8].In this paper we consider the construction of quantum circuits that minimize the use of CNOT gates. Such optimal constructions are of significant interest at least in two contexts. First, they play a role in determining the algorithmic complexity of a given quantum computation, that is, the number of elementary gates required to implement the corresponding n-qubit unitary evolution. A most remarkable result of [2] is the explicit decomposition of an arbitrary U ∈ U(2 n ) as a sequence of CNOT and single-qubit gates. This general construction, however, unavoidably requires exp(n) CNOT gates, which renders the resulting quantum circuit inefficient. We recall, on the other hand, that the n-qubit unitary transformations relevant for quantum computation are precisely those that can be decomposed into only poly(n) elementary gates for large n. Thus, given a unitary transformation U ∈ U(2 n ) of interest, it is important to know how many CNOT gates are required to implement it.Algorithmic complexity is typically concerned with gates involving a large number of qubits. But quantum circuits that minimize the use of CNOT gates are also of interest for gates involving only a reduced number of qubits, for a very practical reason: in present day experiments, two-qubit gates as the CNOT gate are imperfect due to technological limitations. Therefore, in order to minimize the probability that an error occurs in performing a certain unitary evolution U ∈ U(2 n ), it is instrumental that the number of times the qubits interact is as small as possible. Unfortunately, it is not known in general how to optimally decompose U into CNOT and single-qubit gates, not even for a small ...